Shear-driven and channel flow of a liquid film over a corrugated or indented wall

The shape of the interface between two superimposed layers in a two-dimensional channel confined between a planar and a corrugated or indented wall is investigated in the limit of Stokes flow. A perturbation analysis for walls with small-amplitude sinusoidal corrugations reveals that an insoluble surfactant amplifies the deformation of the interface and causes a negative drift in the phase shift under most conditions. The effect is most significant at moderate capillary numbers and for corrugations whose wavelength is large compared to the thickness of the adjacent layer lining the wavy wall. The precise effect of the surfactant depends on the ratio of the fluid viscosities, proximity of the interface to the planar wall, capillary number, and wavelength of the corrugations. When the interface is near the plane wall, introducing surfactant reduces the interfacial amplitude and causes a positive phase shift with respect to the wavy wall. As the interface further approaches the plane wall, the interfacial wave tends to become in phase with the wavy wall, reflecting its unshifted topography. In the second part of this study, a boundary integral method is implemented to compute Stokes flow over a wall with an arbitrary periodic profile, and results are presented for sinusoidal walls and planar walls containing a periodic sequence of square and circular depressions or projections. The numerical results reveal that the linear perturbation theory overestimates the deformation of the interface over a wavy wall, and illustrate the nature of shear-driven film flow over a planar wall with indented topography.